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In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change.
More rigorously, the gradient of a function from the Euclidean space Rn to R is the best linear approximation to that function at any particular point in Rn. To that extent, the gradient is a particular case of the Jacobian.
- Consider a room in which the temperature is given by a scalar field φ, so at each point (x,y,z) the temperature is φ(x,y,z). We will assume that the temperature does not change in time. Then, at each point in the room, the gradient at that point will show the direction in which it gets hottest. The magnitude of the gradient will tell how fast it gets hot in that direction.
- Consider a hill whose height at a point (x,y) is H(x,y). The gradient of H at a point will show the direction of the steepest slope at that point. The magnitude of the gradient will tell how steep the slope actually is. The gradient at a point is perpendicular to the level set going through that point, that is, to the curve of constant height at that point.
The gradient is noted by:
In 3 dimensions, the expression expands to
For example, the gradient of the function φ = 2x + 3y2 - sin(z) is:
The gradient on manifolds
where ξf is the function that takes any point p to the directional derivative of f in the direction ξ evaluated at p. In other words, under some coordinate chart, ξf(p) will be:
- Partial derivation
- Vector calculus
- Nabla in cylindrical and spherical coordinates
- Ion gradient
- Gradient descent
- Level set
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